This invention generally relates to integration of plural sources of information in inversion, and more specifically, to incorporation of plural sources of data of different modalities in inversion. An embodiment of the invention relates to combining such plural sources of data using adjoint based optimization methods.
In many situations, it is essential to be able to predict how the state and the attributes of a system vary over time. For example, prediction of flow of fluids through a reservoir for certain scenarios, e.g. well placement, production optimization, etc., is desirable for supporting business/investment decisions. For efficient recovery of oil and gas from a reservoir, a good understanding of the subsurface attributes and its constitutes is vital.
Conventionally, production data comprised of measurements of pressures in the wells, along with fluid (oil and water) and gas flow rates, is used in attempt to recover the subsurface attributes. The process in which this is performed is called history matching. In this process, the model parameters (such as permeability, porosity, skin, seal factors) are being adjusted so that simulation of flow would match the recorded production data at the wells. There are several strategies for updating the model parameters, including manual trial and error. The most widely accepted approach is based upon non-linear optimization. In this approach, the problem is cast as minimization of an objective function that comprises a measure of the misfit (likelihood) between the actual measured data and the one that is simulated based upon a choice of model parameters. e.g.:
                                          m            ^                    =                                                    arg                ⁢                                                                  ⁢                min                ⁢                                            m                        ⁢                                                          ⁢                                  (                                                            V                      ⁡                                              (                                                  u                          ⁡                                                      (                                                          m                              ;                              y                                                        )                                                                          )                                                              ,                                          d                      ⁡                                              (                        y                        )                                                                              )                                                            ︸                                  data                  ⁢                                                                          ⁢                  misfit                                                                    ⁢                                  ⁢                              s            .            t            .                                                  ⁢                          g              ⁡                              (                                  m                  ,                                      u                    ;                    y                                                  )                                              =          0                ⁢                                  ⁢        constraints                                        where m denotes the model parameters,  stands for a noise/distance model, V is a function that converts the state u (saturation and pressure for flow in porous medium) into simulated measurement, y denotes the experimental design setup and d denotes the real data. As a constraint, the state u must comply with the governing physics of the problem (e.g. flow in porous medium represented by a set of partial differential equations along with appropriate boundary conditions) as represented by the operator g. This objective function may involve additional terms, such as regularization, or additional constraints (e.g. positivity or bounds for some parameters).
Among the various computational methods to solve this optimization problem, adjoint (sensitivity) based methods are acknowledged in the optimization community as superior, and for large-scale problems, often these approaches are the only computationally tractable resort.
Unfortunately, the acquired production data do not convey sufficient information for a complete and stable recovery of the subsurface attributes.